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sum of the momenta which the impinging particles lose in a second gives the amount of the friction which the surfacelayer of the gas has experienced at the wall.

But to form this sum we have to make a hypothetical assumption as to the magnitude of the loss which a single particle suffers on impact at the wall. In the case carried out in § 11 we supposed that no kinetic energy was lost on the collision, but that the particle was reflected from the wall with the same speed as that with which it struck it. We could not now consistently suppose that the particles of gas lose any of their speed; but we might assert that of the forward motion, which all the molecules in common possess, a part must be transformed into heat-motion in consequence of the divergency of the directions in which the impinging particles are reflected. How large a part this will be depends on the degree of unevenness of the surface, with respect to which, therefore, we have to form a definite conception.

The fixed wall on which the particles impinge does not form a plane or continuously curved surface at all; it is made up itself of molecules which leave spaces between each other of sufficient size to allow other molecules to penetrate into them. On the breadth of these molecular pores rests the capacity of solid bodies for condensing1 on their surfaces considerable quantities of gases and vapours, that is, for depriving them of the mobility proper to their state of aggregation. The gaseous molecules penetrate thereby deeply into the interior of the solid body, so that they are able to pass through the walls of glowing tubes 2 whose briskly moving molecules may often leave wide interspaces, and also, when helped by the force of electrolysis, through platinum foil.3

From such observations we are forced to look upon the surfaces of solid bodies, even if most excellently polished, as

Or adsorbing them, according to the modern nomenclature. Compare the observations of Bunsen and Kayser, Wied. Ann. 1883-5.

2 H. Sainte Claire-Deville and Troost, Comptes Rendus, 1863, lvii. p. 965; Pogg. Ann. 1864, cxxii. p. 331.

* Helmholtz, 'Bericht über Versuche des Herrn Dr. Elihu Root,' Monatsber. d. Berl. Akad. 1876, p. 217.

so rough and uneven that a regular flow immediately over them is scarcely even conceivable. The forward motion of the gas becomes almost entirely annihilated, so that we are justified in looking on the external friction as infinitely great and in putting the slip equal to zero, as was in general done in the older investigations on the friction of gases.

In the limiting conceivable case, in which the mean motion of all the particles of gas close by the wall is zero, we must assume that the velocity of flow which those particles have that are coming towards the wall is entirely taken up by those which are coming from it; each particle, therefore, which meets the wall must not only lose on impact its share of the general velocity of flow, but return with an equal component of velocity in the opposite direction. The loss which it has experienced by the impact would, therefore, in the case considered, amount to double the velocity of forward flow.

In reality the loss of velocity will probably be less. I represent it then by Bu, where v denotes the mean velocity of flow and a constant whose value lies between 0 and 2. By the impact of a particle of gas of mass m against the wall, the amount of momentum in the gas is diminished by ßmv. The whole lessening of the momentum in unit time is obtained from this by multiplying it by the number of particles which strike the wall in this time.

In the determination of the pressure by summation of the kinetic energy of all the impacts we found in §§ 11 and 12, by the method first employed by Joule, that the number of particles which strike unit area of the wall in unit time is

NG, where, as before, N is the number of particles in unit volume and G is a mean value of the speed. We cannot, without further consideration, apply this to the case under consideration, because the method there employed is strictly admissible only for the calculation of the kinetic energy, and not of other magnitudes. This value, therefore, for the number of impinging particles is only approximately correct, and for accuracy we must replace it by the number NO calculated in § 37 and § 41* of the Mathematical Appendices, which

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differs but inconsiderably from it, and involves instead of G the smaller arithmetical mean 2 of the molecular speeds.

Multiplication of this number into the loss of momentum of a molecule as found above gives

BmNQv = εv

for the total loss of momentum experienced per unit area per unit time by a gas which flows along a solid wall with the velocity v; or, more shortly expressed, this expression gives the friction of the gas per unit area per unit time on a solid body.

The coefficient which comes into the formula, viz.

ε = 1BmNQ,

is the constant of the external friction of the gas; the formula shows that the theory is in agreement with the law, mentioned already in § 81, which Kundt and Warburg deduced from their observations, viz. that the external friction is proportional to the density p = mN.

For the coefficient of slip we have

y= n/ε = 0·30967 L/‡ß,

= 1.23868 L/B,

which is therefore simply proportional to the free path of the molecules; in denser gases, accordingly, as experiment has proved, the slip on a solid surface is vanishingly small, and it can in general be shown and measured only in rarefied gases.

83. Comparison of the Theory with Experiment

The observations of Kundt and Warburg confirm most excellently the law that immediately follows from the foregoing formula, viz. that, just like L, is inversely proportional to the density and the pressure of the gas. I forbear citing here in fulness the series of numbers given by them, and limit myself to a setting forth of their conclusions.

From every three or four observations under different pressures made with the same arrangement of apparatus they

have deduced a magnitude, denoted by a, whose relation to the coefficient of slip is given by

aD/= 2 × 760,

where D denotes the distance in centimetres between the discs of the apparatus employed with Maxwell's method. By means of this formula I have calculated the following values of the coefficient of slip from the numbers given on pp. 544 and 545 of the memoir cited :

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In this table all the numbers represent centimetres.

In the last column of the table I have put the values of the free paths already obtained in §§ 78 and 79 from the observations on internal friction, and comparison of these with the values of , in the third column, shows in all these cases that the coefficient of slip is nearly equal to the free path, or

5= L.

From this we obtain for the coefficient B, which was introduced as in some sort a measure of the roughness of the surface where the friction takes place,

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By means of this assumption, which corresponds with the theoretical requirement 0 < 8< 2, complete agreement between theory and experiment is obtained.

Later experiments, undertaken by Warburg1 to measure the external friction by experiments on transpiration also, confirm this behaviour, but gave smaller values for the coefficient of slip, and therefore larger values for B. The numbers lately found by Breitenbach,2 on the contrary, agree very well with those given above.

1 Pogg. Ann. 1876, clix. p. 399.

2 Wied. Ann. 1899, lxvii. p. 826.

84. Phenomena in very Rarefied Gases

The molecular free path increases with increase of rarefaction in the ratio of the increase of the volume; so too, therefore, does the coefficient of slip. But we cannot, therefore, believe that the molecular paths in excessively rarefied gases, as in the vacuum of a mercury air-pump, attain a considerable length. If we assume, for instance, that such a pump were to rarefy the air 100,000 times, or to a pressure of less than 10 mm., the free path, which is 0.00001 cm. under atmospheric pressure, would become 1 cm.; it therefore always remains a remarkably small path for a body projected with a speed of something like 450 metres per second. The number of encounters to which a molecule is exposed remains still very great even in such a condition of rarefaction; it would amount to 46,500 per second.

This calculation certainly does not deserve unconditional confidence, if only because Boyle's law does not hold at such small pressures. But, under any circumstances, this consideration is well suited to show that a gas, even if exceedingly rarefied, is anything but a vacuum. A nominally vacuous space, obtained either by an air-pump or even by Torricelli's method, is distinctly not vacuous, but is so uniformly filled with a medium, of a density certainly very small, that our observations will still give us the impression of the space being continuously filled.

The lengthening of the free path helps, indeed, to remove more quickly and easily any inequalities that exist in the distribution of pressure, temperature, &c. According to the kinetic theory, the transference of any action is the result of the transference of molecules from one layer to another. The longer the paths of the molecules, the wider will therefore be the ranges within which all inequalities will be removed.

This remark remains of force, even when the inequality consists in the distribution of electrical tension. This is the reason why electrical discharges in regions of rarefied air, as, for instance, in Geissler's tubes, take place at far greater distances than in denser air.

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